Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$. Is it true that there exists an infinite dimensional reflexive subspace $E$ of $C[0,1]$ s.t. $f(K\cap E)$ is bounded ?

If the answer is affirmative, this would be a very weak kind of Weierstrass-type theorem [and also a very general one, due to the "universality" of $C[0,1]$ (i.e., the Banach-Mazur Embedding Theorem)].

Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$. Is it true that there exists an infinite dimensional reflexive subspace $E$ of $C[0,1]$ s.t. $f(K\cap E)$ is bounded ?

If the answer is affirmative, this would be a very weak kind of Weierstrass-type theorem [and also a very general one, due to the "universality" of $C[0,1]$ (i.e., the Banach-Mazur Embedding Theorem)].

One may also replace $C[0,1]$ by $B[0,1]$, the space of all bounded functions on $[0,1]$, endowed with the sup-norm.